Solving equations is a core part of algebra, but things can get a bit more intimidating once fractions are involved. The good news is that there are several reliable methods to “clear” these fractions and make the equations much easier to handle.
In this guide, we look at four different problems, ranging from simple linear equations to complex fractional expressions.
1. Dealing with Single Denominators
In our first example, we look at a classic linear equation with fractions on both sides: ** (2x / 3) – 1 = x / 2 **
Instead of finding a Lowest Common Multiple (LCM) immediately, a great way to start is by removing one denominator at a time. By multiplying the entire equation by the first denominator and then the second, you can systematically simplify the problem into a standard equation.
Watch the step-by-step solution here: https://youtu.be/fDtaEcDbOXQ
2. Fractions and Brackets
When you have fractions outside of brackets, such as: ** 3/4 (x – 1) = 1/3 (2x – 1) **
A highly effective method is to multiply both sides of the equation by a number that both denominators go into (in this case, 12).
The key here is mental arithmetic: work out what 3/4 of 12 is and what 1/3 of 12 is. Once you have those whole numbers, you can expand the brackets and solve for x without having to worry about fractions for the rest of the calculation.
See the full walkthrough here: https://youtu.be/YNy4a0JtzQU
3. Advanced Algebraic Fractions (Part A)
As questions get more challenging, you will often see multiple terms in the numerator, such as: ** ((x + 1) / 7) – (3(x – 2) / 14) = 1 **
In this scenario, we use the LCM (14) to clear the denominators. The most important thing to watch out for here is the minus sign between the two fractions. It is vital to treat the second numerator as being inside a bracket to ensure you distribute the negative sign correctly across both terms.
4. Complex Numerators and Negatives (Part B)
Finally, we look at equations where the variable ‘x’ is subtracted within the numerator: ** ((6 – 3x) / 3) – ((5x + 12) / 4) = -1 **
By multiplying by the LCM of 12, we can remove the fractions. However, accuracy is key here. Expanding the numerators and carefully collecting like terms is the only way to reach the correct final value. This problem is a brilliant test of your algebraic fluency.
Watch the solutions for Questions 3 and 4 here: https://youtu.be/V-74iBGL8J4
Top Tips for Success
- Multiply every term: If you multiply one part of the equation by a number, you must multiply every single term (including constants like ‘1’ or ‘-1’).
- Use Brackets: When removing a denominator from a fraction with multiple terms on top, put that numerator in brackets to protect the signs.
- Check your answer: Always substitute your final value of x back into the original equation to see if the left side equals the right side.